Bring the fish to room temperature for 10–15 minutes before cooking. This ensures even cooking and prevents the exterior from overcooking while the center comes up to temperature.

Pat the fillets dry very well with paper towels and season both sides with 1 tsp salt, 1/2 tsp black pepper, and the paprika. Drying the surface removes moisture so the fish browns rather than steams; seasoning early lets flavors penetrate the flesh.

Heat a large stainless-steel or cast-iron skillet over medium-high heat until hot (about 2 minutes). Add 2 tablespoons olive oil and swirl to coat. A hot pan and a little oil produce a Maillard crust that gives the fish color and flavor.

Place the fillets presentation-side down (skin-side if present) in the pan, spacing them so they don’t touch. Press gently for 6–8 seconds to ensure full contact. Leave undisturbed for 3–4 minutes until a deep golden crust forms. Don’t flip too early — let the crust develop to lock in juices.

Flip the fillets and immediately reduce heat to medium. Cook the second side 2–4 minutes depending on thickness: the fish is done when it flakes easily and the internal temperature reaches 60–63°C (140–145°F). Lowering heat after flipping prevents the butter from burning while finishing the fish.

While the fish cooks, in a separate small skillet melt 2 tablespoons of the butter over medium heat. Add the thinly sliced onion and a pinch of salt; cook, stirring occasionally, until deeply golden and caramelized — about 8–10 minutes. Caramelizing transforms the onion's sharpness into sweet, savory depth that complements the fish.

When the onions are nearly caramelized, push them to the side and add the remaining 2 tablespoons butter; let it melt and foam. Add the garlic and dried thyme, and cook for 30–45 seconds until fragrant. Watch carefully — garlic browns quickly. Browning the butter and garlic gives a nutty, complex base for the sauce.

Tilt the pan and, using a spoon, baste the butter and onion mixture over the finished fillets for 20–30 seconds, then transfer fish to warm plates. Basting coats the fish in glossy flavor and keeps it moist.

Spoon the caramelized onion and brown-butter sauce over the fillets, sprinkle the remaining 1/2 tsp salt and 1/2 tsp black pepper to taste, and let rest 1–2 minutes. Resting allows residual heat to finish the fish and for flavors to meld.

Serve immediately. Suggested plating: place a fillet in the center, top with a spoonful of onions and sauce, and wipe any burnt butter from the rim of the plate for a clean presentation.